On generating sets of infinite symmetric group

TL;DR

This paper investigates generating sets within the infinite symmetric group, establishing conditions and criteria for when certain families of subsets can generate the entire group.

Contribution

It introduces new criteria for generating sets in the infinite symmetric group based on eigenspace cardinalities.

Findings

Identified families of subsets that generate the infinite symmetric group

Derived a criterion for when these subsets generate the group

Enhanced understanding of the structure of generating sets in infinite groups

Abstract

It was shown that in a group of bijections of an infinite set some families of subsets, related to the cardinality of some eigenspaces, are generating. Besides, we derived a criterion for generating by sets of this kind.